Average Error: 0.0 → 0.0
Time: 8.7s
Precision: 64
Internal Precision: 320
\[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]
↓
\[\frac{\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y}{2}\]

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[\Re(\left(\frac{e^{x} + e^{-x}}{2} \cdot \cos y + \frac{e^{x} - e^{-x}}{2} \cdot \sin y i\right))\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\frac{(\left(\cos y\right) \cdot \left(e^{x}\right) + \left(\frac{\cos y}{e^{x}}\right))_*}{2}}\]
  3. Using strategy rm

  4. Applied fma-udef0.0

    \[\leadsto \frac{\color{blue}{\cos y \cdot e^{x} + \frac{\cos y}{e^{x}}}}{2}\]

  5. Final simplification0.0

    \[\leadsto \frac{\frac{\cos y}{e^{x}} + e^{x} \cdot \cos y}{2}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (x y)
  :name "Euler formula real part (p55)"
  (re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y)))))

Details

Time bar (total: 8.1s)Debug log

sample158.0ms

Algorithm
halfpoints

simplify101.0ms

Counts
1 → 1
Calls

1 calls. Slowest were:

101.0ms
(re (complex (* (/ (+ (exp x) (exp (- x))) 2) (cos y)) (* (/ (- (exp x) (exp (- x))) 2) (sin y))))

prune17.0ms

Pruning

1 alts after pruning (1 fresh and 0 done)

Merged error: 0.0b

localize40.0ms

Local error

Found 2 expressions with local error:

0.0b
(fma (cos y) (exp x) (/ (cos y) (exp x)))
0.0b
(/ (cos y) (exp x))

rewrite8.0ms

Algorithm
rewrite-expression-head
Counts
2 → 39
Calls

2 calls. Slowest were:

7.0ms
(/ (cos y) (exp x))
0.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))

series165.0ms

Counts
2 → 6
Calls

2 calls. Slowest were:

112.0ms
(fma (cos y) (exp x) (/ (cos y) (exp x)))
53.0ms
(/ (cos y) (exp x))

simplify292.0ms

Counts
31 → 45
Calls

31 calls. Slowest were:

110.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
74.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
21.0ms
(/ (* (cbrt (cos y)) (cbrt (cos y))) (* (cbrt (exp x)) (cbrt (exp x))))

prune423.0ms

Pruning

2 alts after pruning (1 fresh and 1 done)

Merged error: 0b

localize10.0ms

Local error

Found 3 expressions with local error:

0.0b
(+ (* (cos y) (exp x)) (/ (cos y) (exp x)))
0.0b
(* (cos y) (exp x))
0.0b
(/ (cos y) (exp x))

rewrite22.0ms

Algorithm
rewrite-expression-head
Counts
3 → 63
Calls

3 calls. Slowest were:

8.0ms
(+ (* (cos y) (exp x)) (/ (cos y) (exp x)))
7.0ms
(/ (cos y) (exp x))
6.0ms
(* (cos y) (exp x))

series135.0ms

Counts
3 → 9
Calls

3 calls. Slowest were:

64.0ms
(+ (* (cos y) (exp x)) (/ (cos y) (exp x)))
36.0ms
(* (cos y) (exp x))
34.0ms
(/ (cos y) (exp x))

simplify417.0ms

Counts
46 → 72
Calls

46 calls. Slowest were:

103.0ms
(/ (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))
80.0ms
(- 1 (+ (* 1/2 (pow y 2)) x))
54.0ms
(* (* (* (cos y) (cos y)) (cos y)) (* (* (exp x) (exp x)) (exp x)))

prune816.0ms

Pruning

2 alts after pruning (0 fresh and 2 done)

Merged error: 0b

regimes52.0ms

Accuracy

0% (0.0b remaining)

Error of 0.0b against oracle of 0.0b and baseline of 0.0b

bsearch1.0ms

end0.0ms

sample5.4s

Algorithm
halfpoints